A couple of things to add to the other members' excellent explanations:
'smart' is used less in the UK than it is in the USA. The main meaning of 'smart' in British English refers to dress and appearance, in contrast to casual. (eg 'You need to dress smartly for your job interview - make sure you put on a smart suit).
'smart' is sometimes used in Britain the American sense of 'clever', particularly in set phrases such as 'That was a smart move!' 'He's a pretty smart businessman' or 'Don't give me any of your smart answers!' . In this last example, it has a negative implication ie rude and cheeky.
As the other members have said, a clever person is good at working things out for themselves, often to their own advantage. Intelligent is somehow deeper, a reference to your brain power rather than your behaviour.
And, again as others have said, 'bright' is a term that adults use for children and young people. So it wouldn't be appropriate to use this for one of your colleagues, for example - it would sound condescending.